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Cofinality and Measurability of the First Three Uncountable Cardinals

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Cofinality and Measurability of the First Three Uncountable Cardinals COFINALITY AND MEASURABILITY OF THE FIRST THREE UNCOUNTABLE CARDINALS ARTHUR W. APTER, STEPHEN C. JACKSON, BENEDIKT LOWEÄ Abstract. This paper discusses models of set theory without the Axiom of Choice. We investigate all possible patterns of the co¯nality function and the distribution of measurability on the ¯rst three uncountable cardinals. The result relies heavily on a strengthening of an unpublished result of Kechris: we prove (under AD) that there is a cardinal · such that the triple (·; ·+;·++) satis¯es the strong polarized partition property. 1. Introduction In ZFC, small cardinals such as @1, @2, and @3 cannot be measurable, as mea- surability implies strong inaccessibility; they cannot be singular either, as successor cardinals are always regular. So, in ZFC, these three cardinals are non-measurable regular cardinals. But both of the mentioned results use the Axiom of Choice, and there are many known situations in set theory where these small cardinals are either singular or measurable: in the Feferman-L¶evymodel, @1 has countable co¯nality (cf. [Jec03, Example 15.57]), in the model constructed independently by Jech and Takeuti, @1 is measurable (cf. [Jec03, Theorem 21.16]), and in models of AD, both @1 and @2 are measurable and cf(@3) = @2 (cf. [Kan94, Theorem 28.2, Theorem 28.6, and Corollary 28.8]). Simple adaptations of the Feferman-L¶evyand Jech and Takeuti arguments show that one can also make @2 or @3 singular or measurable, but is it possible to control these properties simultaneously for the three cardinals @1, @2 and @3? In this paper, we investigate all possible patterns of measurability and co¯nality for the three mentioned cardinals. Combinatorially, there are exactly 60 (= 3 £ 4 £ 5) such patterns: @1 can be measurable, regular non-measurable, or singular (3 possibilities); @2 can be measurable, regular non-measurable, or have co¯nality @1 or @0 (4 possibilities); and @3 can be measurable, regular non-measurable, or have co¯nality @2, @1, or @0 (5 possibilities). Out of these 60 patterns, 13 are impossible for trivial reasons: for instance, if @1 is singular, then @2 cannot have co¯nality @1, as co¯nalities always must be regular cardinals. Date: May 4, 2009. 2000 Mathematics Subject Classi¯cation. Primary 03E02, 03E35, 03E55, 03E60; secondary 03E10, 03E15, 03E25. The ¯rst author's visits to Amsterdam and the third author's visit to New York were partially supported by the DFG-NWO Bilateral Grant KO 1353-5/1/DN 62-630. In addition, the ¯rst au- thor wishes to acknowledge the support of various PSC-CUNY and CUNY Collaborative Incentive grants. The third author would like to thank the CUNY Graduate Center Mathematics Program for their hospitality and partial ¯nancial support during his stay in New York. 1 For these patterns, we shall use the labels M and @n, standing for \measurable" and \non-measurable and co¯nality @n", respectively, and write [ x1 = x2 = x3 ] for the statement \@1 has property x1, @2 has property x2, and @3 has property x3". For technical reasons, we shall assume that all measurable cardinals carry a normal ultra¯lter. In all of our constructions of models, this will be the case; this slightly non-standard de¯nition does make a di®erence for lower bounds. We discuss this in somewhat more detail in x9. We shall go through all 60 possible patterns and prove 47 of them to be consistent from the appropriate large cardinal axioms, and 13 of them to be inconsistent. Instead of proving a ridiculously large number of results, we have arranged the paper as follows: in x 2 we shall provide some basic tools for forcing in the ZF context, some of them using polarized partition properties. These tools will allow us to look at the 47 consistent patterns as a graph reducing all cases to eight base cases in x 3. In xx 4 and 5, we prove the consistency of all base cases. In the former section, we use techniques from forcing and large cardinals; in the latter, we rely on AD (and in particular, on certain polarized partition properties that hold under AD). The main polarized partition property we use is a generalization of an unpub- lished theorem of Kechris from the 1980s (cf. [AH86, p. 600]). In this paper, we give a proof of (a slightly stronger version of) Kechris' result (x 6) and generalize it to higher exponents (xx 7 and 8), as needed in our applications in x 5. Finally, in x 9, we shall summarize upper and lower consistency strength bounds of all 60 patterns. 2. A Toolkit In this section, we list a number of basic forcing facts, the main de¯nitions of polarized partition properties and some facts about AD on which our analysis rests. 2.1. Forcing facts. Theorem 1. If V j= ZF+\· is a measurable cardinal", P· is P·r¶³kr¶yforcing for ·, and G is P·-generic over V, then in the generic extension V[G], cf(·) = @0, any cardinal having co¯nality · in V now has co¯nality @0, and the co¯nalities and measurability of all other cardinals are unchanged. Proof. This is [Apt96, Lemmas 1.2, 1.3 and 1.5]. q.e.d. Theorem 2. If V j= ZF+\@1 is measurable", Add(!; !1) is the partial order for adding !1 many Cohen reals, and G is Add(!; !1)-generic over V, then in the generic extension V[G], @1 is regular but non-measurable and co¯nalities and measurability of all other cardinals are unchanged. Proof. The fact that @1 becomes non-measurable is a special case of the general ZF-result (due to Ulam) that if · injects into 2¸ for some ¸ < ·, then there cannot be a ·-complete ultra¯lter on · (cf. [Kan94, Theorem 2.8]). Obviously, the !1- ! sequence of Cohen reals produces an injection of !1 into 2 . Since Add(!; !1) is canonically well-orderable and jAdd(!; !1)j = @1, the proof that all cardinals and 2 co¯nalities are preserved is the same as when AC is true. Since jAdd(!; !1)j = @1, the argument given in the proof of [AH86, Lemma 2.1] shows that the measurability of all cardinals greater than @1 is preserved. q.e.d. Theorem 3. If V j= ZFC+\· < ¸ are measurable cardinals", then for x3 2 f@0; @1; @2; @3; Mg, there is a symmetric submodel Nx3 satisfying [ M = @2 = x3 ]. If x3 6= M, only one measurable cardinal is needed in V. Proof. We sketch the proof of Theorem 3. Without loss of generality, we assume that GCH holds in V. Let G0 be Col(!; <·)-generic over V, where for ½ < ³, ½ a regular cardinal, ³ a cardinal, Col(½; <³) is the L¶evycollapse of all cardinals less than ³ to ½. For H which is Col(½; <³)-generic over V and » 2 (½; ³) a cardinal, let + H¹» be all elements of H which are members of Col(½; <»). Let G1 be Col(· ; <γ)- generic over V, where γ is either ·+!, ·+·, ·+´ for ´ = ·+, or ¸. We write HDV(X) for the class of sets hereditarily V-de¯nable with a parameter from X. Consider the symmetric model Nx3 := HDV(fG0¹± ; ± 2 (!; ·) and ± is a cardinalg[ + + fG1¹± ; ± 2 (· ; γ) and ± is a cardinalg). Since in V, there is a · sequence of subsets of ·, standard arguments show that Nx3 is a model for [ M = @2 = @0 ], +! +· +´ [ M = @2 = @1 ], [ M = @2 = @2 ], or [ M = @2 = M ], for γ either · , · , · for ´ = ·+, or ¸ respectively. Since in V, there is a ·+ sequence of subsets of · and a ·++ + sequence of subsets of · , Nx3 := HDV(fG0¹± ; ± 2 (!; ·) and ± is a cardinalg) is a model for [ M = @2 = @3 ]. Clearly, the only time a second measurable cardinal is needed in the construction is for the pattern [ M = @2 = M ]. q.e.d. Theorem 4. Suppose i 2 !. Let V j= ZF + \· is a limit cardinal" + \¸ := ·+i". Let G be Col(!; <·)-generic over V. Consider the model M obtained by symmetrically collapsing · to @1, i.e., the model M := HDV(fG¹± ; ± 2 (!; ·) and ± is a cardinalg). Then the following hold: (i) If V j= \¸ is measurable", then M j=\¸ = @i+1 is measurable". +j (ii) If V j= \cf(¸) = · for some j · i", then M j= \cf(¸) = @j+1". Proof. Since V j= \Col(!; <·) is canonically well-orderable and jCol(!; <·)j = ·", (i) follows from the argument given in the proof of [AH86, Lemma 2.1], the fact that · = @1 in M, and the fact that cardinals at and above · are preserved to M; (ii) follows from the fact that · = @1 in M and the fact that cardinals and co¯nalities at and above · are preserved to M. q.e.d. 2.2. Polarized partition properties. Fix a strictly increasing triple (·0;·1;·2) of cardinals and an ordinal ± · ·0. We say a function f : 3 £ ± ! On is a block function if ·i¡1 < f(i; ®) < ·i for i 2 3 (and ·¡1 := 0), and we say it is increasing if f(i; ®) < f(i; ¯) whenever ® < ¯. We denote the set of increasing block functions by IBF±. If H~ = (H0;H1;H2) is a tuple such that Hi ⊆ ·i (for i 2 3), we de¯ne a subset FH~ ⊆ IBF± by f 2 FH;±~ : () for all ® 2 ± and i 2 3, we have f(i; ®) 2 Hi: If P ⊆ IBF± is a partition of all increasing block functions into two disjoint sets, ~ we call a triple H ±-homogeneous for P if either FH;±~ ⊆ P or FH;±~ \ P = ?. 3 S For Ramsey-type partition properties, we also de¯ne the set IBF<± := ®<± IBF®, and for a partition P ⊆ IBF<±, we say that a tuple H~ is <±-homogeneous for P if for all ® < ±, either FH;®~ ⊆ P or FH;®~ \ P = ?.
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